×

zbMATH — the first resource for mathematics

Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. (English) Zbl 1282.60096
The paper addresses a noninteracting unbounded spin system with the mean spin conservation constraint. For the canonical ensemble, with the choice of super-quadratic single-site potentials, an old Varadhan’s problem [S. R. S. Varadhan, Pitman Res. Notes Math. Ser. 283, 75–128 (1993; Zbl 0793.60105)] is addressed of the validity fo the spectral gap inequality and next its strenghthening by means of the logarithmic Sobolev inequality. Since standard criterions to this end (e.g. the tensorization principle, Bakry-Émery and Holley-Stroock criterions) fail for the canonical ensemble, new techniques are developed. They stem from the previous (co-authored by F. Otto) work on the two-scale approach [N. Grunewald et al., Ann. Inst. Henri Poincaré, Probab. Stat. 45, No. 2, 302–351 (2009; Zbl 1179.60068)]. Using the asymmetric Brascamp-Lieb type inequality for covariances, the task of deriving a unform logarithmic Sobolev inequality is reduced to the convexification of the coarse grained Hamiltonian by iterated renormalization. A local Cramer theorem is employed, the idea bororowed from [N. Grunewald et al., loc. cit.]

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J25 Continuous-time Markov processes on general state spaces
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Bakry, D. and Émery, M. (1985). Diffusions hypercontractives. In Séminaire de Probabilités , XIX , 1983 / 84. Lecture Notes in Math. 1123 177-206. Springer, Berlin. · Zbl 0561.60080 · doi:10.1007/BFb0075847 · numdam:SPS_1985__19__177_0 · eudml:113511
[2] Barthe, F. and Wolff, P. (2009). Remarks on non-interacting conservative spin systems: The case of gamma distributions. Stochastic Process. Appl. 119 2711-2723. · Zbl 1169.60325 · doi:10.1016/j.spa.2009.02.004
[3] Bobkov, S. G. (1999). Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 1903-1921. · Zbl 0964.60013 · doi:10.1214/aop/1022677553
[4] Brascamp, H. J. and Lieb, E. H. (1976). On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 366-389. · Zbl 0334.26009 · doi:10.1016/0022-1236(76)90004-5
[5] Caputo, P. (2003). Uniform Poincaré inequalities for unbounded conservative spin systems: The non-interacting case. Stochastic Process. Appl. 106 223-244. · Zbl 1075.60581 · doi:10.1016/S0304-4149(03)00044-9
[6] Chafaï, D. (2003). Glauber versus Kawasaki for spectral gap and logarithmic Sobolev inequalities of some unbounded conservative spin systems. Markov Process. Related Fields 9 341-362. · Zbl 1040.60081
[7] Csiszár, I. (1967). Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungar. 2 299-318. · Zbl 0157.25802
[8] Evans, L. C. and Gariepy, R. F. (1992). Measure Theory and Fine Properties of Functions . CRC Press, Boca Raton, FL. · Zbl 0804.28001
[9] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II , 2nd ed. Wiley, New York. · Zbl 0219.60003
[10] Gross, L. (1975). Logarithmic Sobolev inequalities. Amer. J. Math. 97 1061-1083. · Zbl 0318.46049 · doi:10.2307/2373688
[11] Grunewald, N., Otto, F., Villani, C. and Westdickenberg, M. G. (2009). A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Ann. Inst. Henri Poincaré Probab. Stat. 45 302-351. · Zbl 1179.60068 · doi:10.1214/07-AIHP200 · eudml:78025
[12] Guionnet, A. and Zegarlinski, B. (2003). Lectures on logarithmic Sobolev inequalities. In Séminaire de Probabilités , XXXVI. Lecture Notes in Math. 1801 1-134. Springer, Berlin. · Zbl 1125.60111 · doi:10.1007/b10068 · numdam:SPS_2002__36__1_0 · eudml:114087
[13] Guo, M. Z., Papanicolaou, G. C. and Varadhan, S. R. S. (1988). Nonlinear diffusion limit for a system with nearest neighbor interactions. Comm. Math. Phys. 118 31-59. · Zbl 0652.60107 · doi:10.1007/BF01218476
[14] Holley, R. and Stroock, D. (1987). Logarithmic Sobolev inequalities and stochastic Ising models. J. Stat. Phys. 46 1159-1194. · Zbl 0682.60109 · doi:10.1007/BF01011161
[15] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 320 . Springer, Berlin. · Zbl 0927.60002
[16] Kullback, S. (1967). A lower bound for discrimination information in terms of variation. IEEE Trans. Inform. 4 126-127.
[17] Landim, C., Panizo, G. and Yau, H. T. (2002). Spectral gap and logarithmic Sobolev inequality for unbounded conservative spin systems. Ann. Inst. Henri Poincaré Probab. Stat. 38 739-777. · Zbl 1022.60087 · doi:10.1016/S0246-0203(02)01108-1 · numdam:AIHPB_2002__38_5_739_0 · eudml:77731
[18] Ledoux, M. (2001). Logarithmic Sobolev inequalities for unbounded spin systems revisited. In Séminaire de Probabilités , XXXV. Lecture Notes in Math. 1755 167-194. Springer, Berlin. · Zbl 0979.60096 · doi:10.1007/978-3-540-44671-2_13 · numdam:SPS_2001__35__167_0 · eudml:114059
[19] Lu, S. L. and Yau, H.-T. (1993). Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Comm. Math. Phys. 156 399-433. · Zbl 0779.60078 · doi:10.1007/BF02098489
[20] Menz, G. (2011). LSI for Kawasaki dynamics with weak interaction. Comm. Math. Phys. 307 817-860. · Zbl 1266.82038 · doi:10.1007/s00220-011-1326-6
[21] Otto, F. and Villani, C. (2000). Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 361-400. · Zbl 0985.58019 · doi:10.1006/jfan.2000.3557
[22] Royer, G. (1999). Une Initiation aux Inégalités de Sobolev Logarithmiques. Cours Spécialisés [ Specialized Courses ] 5 . Société Mathématique de France, Paris. · Zbl 0927.60006
[23] Varadhan, S. R. S. (1993). Nonlinear diffusion limit for a system with nearest neighbor interactions. II. In Asymptotic Problems in Probability Theory : Stochastic Models and Diffusions on Fractals ( Sanda/Kyoto , 1990). Pitman Res. Notes Math. Ser. 283 75-128. Longman Sci. Tech., Harlow. · Zbl 0793.60105
[24] Yau, H.-T. (1996). Logarithmic Sobolev inequality for lattice gases with mixing conditions. Comm. Math. Phys. 181 367-408. · Zbl 0864.60079 · doi:10.1007/BF02101009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.